New minting would clash with any of the 6000 full nodes that did not upgrade.
SegWit would not from my understanding because of "hacks".
New minting of the original _bitcoin-1_ would clash with the non-upgraded full nodes.
A new soft-fork-currency is like bitcoin-2, non-upgraders would not notice it has emerged, they will happily accept them (blocks which contain bitcoin-2) as valid blocks.
Soft forks would not violate the former validation rules, it just makes them more narrow (such as, former validation rules _AND_ blocks must contain bitcoin-2 information).
Actual implementation may not be so beautiful one, depending on the available features/space.